The Worldline Formalism in AdS/CFT

Abstract

This thesis develops the worldline formalism as a semiclassical framework for holographic correlators in AdS/CFT. The worldline approach rewrites bulk amplitudes as path integrals over particle trajectories, making it particularly natural in the regime of heavy external operators with large conformal dimension Δ, where the bulk dynamics is dominated by classical geodesics. We formulate the open worldline path integral for scalar fields in AdSd+1 in Poincar'e coordinates, including the curved-space measure, ghost sector, and time-slicing counterterms, and organize the resulting expressions as an expansion around geodesic saddles. As a first application, we compute the AdS scalar propagator semiclassically and recover the standard bulk-to-boundary propagator together with its first subleading correction in 1/Δ. We then apply the same formalism to heavy-light correlators in bulk ϕ3 and ϕ4 theories. For two heavy operators and N−2 light insertions, we derive the leading worldline representation of the general correlator, and we work out the heavy-heavy-light three-point function and heavy-heavy-light-light four-point functions through first subleading order, matching the corresponding Witten-diagram results. Finally, we discuss how the single-geodesic picture generalizes to joined worldline saddles for correlators with three and four heavy operators. These results show that the worldline proper-time expansion provides a natural and systematic organization of the large-Δ expansion of holographic correlators in AdS.